Let \geq 2$, let ,K'$ be fields such that '$ is a quadratic Galois-extension of $ and let $\theta$ denote the unique nontrivial element in (K'/K)$. Suppose the symplectic dual polar space (2n-1,K)$ is fully and isometrically embedded into the Hermitian dual polar space (2n-1,K',\theta)$. We prove that the projective embedding
of (2n-1,K)$ induced by the Grassmann-embedding of (2n-1,K',\theta)$ is isomorphic to the Grassmann-embedding of (2n-1,K)$. We also prove that if $ is even, then the set of points of (2n-1,K',\theta)$ at distance at most $\frac{n}{2}-1$ from (2n-1,K)$ is a hyperplane of (2n-1,K',\theta)$ which arises from the Grassmann-embedding of (2n-1,K',\theta)$.

@article{879154,
abstract = {Let {\textbackslash}geq 2\$, let ,K'\$ be fields such that '\$ is a quadratic Galois-extension of \$ and let \${\textbackslash}theta\$ denote the unique nontrivial element in (K'/K)\$. Suppose the symplectic dual polar space (2n-1,K)\$ is fully and isometrically embedded into the Hermitian dual polar space (2n-1,K',{\textbackslash}theta)\$. We prove that the projective embedding
of (2n-1,K)\$ induced by the Grassmann-embedding of (2n-1,K',{\textbackslash}theta)\$ is isomorphic to the Grassmann-embedding of (2n-1,K)\$. We also prove that if \$ is even, then the set of points of (2n-1,K',{\textbackslash}theta)\$ at distance at most \${\textbackslash}frac\{n\}\{2\}-1\$ from (2n-1,K)\$ is a hyperplane of (2n-1,K',{\textbackslash}theta)\$ which arises from the Grassmann-embedding of (2n-1,K',{\textbackslash}theta)\$.},
author = {De Bruyn, Bart},
issn = {0024-3795},
journal = {Linear Algebra and its Applications},
keyword = {symplectic/Hermitian dual polar space,hyperplane,Grassmann-embedding},
language = {eng},
pages = {2541--2552},
title = {On isometric full embeddings of symplectic dual polar spaces into Hermitian dual polar spaces.},
url = {http://dx.doi.org/10.1016/j.laa.2008.12.025},
volume = {430},
year = {2009},
}